Hi! My genetic cousin who was adopted, has the same questions as you. I happened to have graduated the Academy of Police in Romania (and earned a law degree), but also have a Master in Law in America (close to obtaining the Bar license). But I'm not an American or Romanian licensed Attorney, so, take this with a pinch of salt.
You should be able to resolve all your legal matters at the closest Romanian Embassy; if they send you to Romania, remind them that as soon as you walked onto that door, you, as a born Romanian citizen, are on the juridical territory of Romania, and they are there to help you. If they give you a hassle, ask to speak with the Counselor (someone who is actually licensed to practice law); I had an ugly experience in DC but fixed it right away as soon as I heard a clerk sending my mother to Romania for a damn power of attorney that could have been done there. Once I spoke with a Counselor, we had peace and left with a resolution; in fact, she gave me her email to contact her any time I would need her (and I did) which created no more issues.
I looked at the Romanian Embassy website in the USA. Unfortunately, I can't find the English version of the page that explains that you can obtain documents from them: Use Google Translate if you need to. This is the page for the birth certificate:
It's not an easy process considering your situation but as a former forensic I can guarantee that you have a way to obtain a Cazier (a sort of a civilian proof of a security clearance that you don't have any criminal record). If you need to return to Romania in the city where you were born, you need to know exactly what paperwork you need to bring (to which the Apostille from the Convention of Hague is affixed). You'll need to translate it from English to Romanian (it's cheaper to do it in Romania, and I can help with guidance) but hopefully you won't get that deep. I have childhood friends from CA who had to return to Romania 3-4 times to resolve an issue because they were not explained the entire process but only in bits and pieces (step by step but as they went through the process, not overall). Pretty annoying and you bet, very expensive.
I'm not sure about your statement of updating the birth certificate; adoption does not change the name of the biological parents on this document. You'll always need the Certificate of Adoption which in many cases is an actual Judge Order/ Decree that has to be translated from the language of origin to the target language (including the Apostille). Unless you refer to any change of names (yours of your biological or adopting parents) in which case it still doesn't affect the birth certificate, I don't see a need for an update. Romania does not apply American rules which in some states change the birth certificate of a person who changed his/ her biological gender. This is a very conservative society.
Don't expect privacy when discussing with the staff! I know it's weird but you can ask for a private place if people are waiting but shouldn't have that issue if you have an appointment. After all, the European Union has the toughest privacy laws in the world, and Romania, as a member, should implement them, too. Stand your ground! Be firm but respectful which is not something that many Romanians would appreciate. I fight this mentality on a daily basis here. Many are like sheep and they don't know their rights which pisses me off.
I hope this helps! Let me know if you have other questions. I could dig for more info for you.
Bella